**2. Methodology**

Following [25,26], *f*(*η*) can be obtained as

$$f(\eta) = \frac{1}{\beta(1 + \gamma\beta - \mu\beta^2)} \left(1 - e^{-\beta\eta}\right),\tag{10}$$

where *β* is the positive root of the following nonlinear equation:

$$
\mu \beta^4 - \gamma \beta^3 - \left(1 + \lambda \mu + \mu M(1 - \phi)^{2.5}\right) \beta^2 + \left(\gamma \lambda + M \gamma (1 - \phi)^{2.5}\right) \beta + \left(\phi\_1 + \lambda + M(1 - \phi)^{2.5}\right) = 0.\tag{11}
$$

On inserting Equation (11) into Equation (9), we obtain the following nonlinear ordinary differential equation (ODE) for *g*(*η*):

$$\left\{\mathbf{g''}\left(\eta\right) + \Omega\left(1 - \varepsilon^{-\beta\eta}\right)\mathbf{g'}\left(\eta\right) - \mathrm{KScg}\left(\eta\right)\left(1 - \mathbf{g}\left(\eta\right)\right)^{2} = 0,\tag{12}$$

where Ω is defined as

$$
\Omega = \frac{S\varepsilon}{\beta(1 + \gamma\beta - \mu\beta^2)}.\tag{13}
$$

The skin friction coefficient is defined in [20] by Equation (14) and hence Equation (15) is obtained by using the exact expression for *f*(*η*) in Equation (10).

$$\text{Skin friction coefficient} = -\frac{2}{\left(1-\phi\right)^{2.5}} f''(0),\tag{14}$$

$$\text{Skin friction coefficient} = \frac{2\beta}{\left(1 - \phi\right)^{2.5} \left(1 + \gamma\beta - \mu\beta^2\right)}.\tag{15}$$

At the special case, *K* → 0, the analytic solution of Equation (12) is given as

$$\mathcal{G}(\eta) = \frac{1 + \varepsilon \Gamma\left(\Omega/\beta, \Omega/\beta e^{-\beta \eta}, \Omega/\beta\right)}{1 + \varepsilon \Gamma(\Omega/\beta, 0, \Omega/\beta)},\tag{16}$$

where *ε* is defined by

$$\varepsilon = K\_s(\beta)^{\Omega/\beta - 1} \left( e^{\beta^2} / \Omega \right)^{\Omega/\beta}. \tag{17}$$

This case may be of a physical meaning when only the heterogenous reactions occur. The concentration is therefore given as

$$\log(0) = \frac{1}{1 + \varepsilon \Gamma(\Omega/\beta, 0, \Omega/\beta)}.\tag{18}$$
